L11n71

L11n70

L11n72

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Link Presentations

[edit Notes on L11n71's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X7,16,8,17 X17,22,18,5 X11,18,12,19 X21,12,22,13 X13,20,14,21 X19,14,20,15 X15,8,16,9 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ -\frac{2 u v^3-u v^2-2 u v+2 u+2 v^3-2 v^2-v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{13} z^{-1} +2 z a^{11}+a^{11} z^{-1} +3 z a^9+2 a^9 z^{-1} -z^5 a^7-5 z^3 a^7-5 z a^7-2 a^7 z^{-1} -z^5 a^5-4 z^3 a^5-2 z a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^8-7 a^{14} z^6+16 a^{14} z^4-14 a^{14} z^2+4 a^{14}+a^{13} z^9-6 a^{13} z^7+10 a^{13} z^5-5 a^{13} z^3+a^{13} z-a^{13} z^{-1} +3 a^{12} z^8-20 a^{12} z^6+41 a^{12} z^4-33 a^{12} z^2+9 a^{12}+a^{11} z^9-5 a^{11} z^7+3 a^{11} z^5+5 a^{11} z^3-2 a^{11} z-a^{11} z^{-1} +2 a^{10} z^8-13 a^{10} z^6+22 a^{10} z^4-13 a^{10} z^2+4 a^{10}+2 a^9 z^7-14 a^9 z^5+24 a^9 z^3-13 a^9 z+2 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+8 a^8 z^2-2 a^8+a^7 z^7-6 a^7 z^5+10 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^6-4 a^6 z^4+2 a^6 z^2+a^5 z^5-4 a^5 z^3+2 a^5 z }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     1 1 0 -8                 1 1     0 -10               1 1 1     1 -12             2 4 1       1 -14           1   1         2 -16         1 3 2           0 -18       2 1               1 -20       1 1               0 -22   1 2                   -1 -24                         0 -26 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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